Metafrontier Frameworks for the Study of Firm-Level Efficiencies and Technology Ratios

Transcription

1 Tle: Metafrontier Framewors for the Study of Firm-Level Efficiencies and Technology Ratios Authors: Associate Professor Christopher J. O Donnell 1 School of Economics Universy of Queensland Brisbane, QLD 4072 Australia c.odonnell@economics.uq.edu.au Professor D.S. Prasada Rao School of Economics Universy of Queensland Brisbane, QLD 4072 Australia Dr. George E. Battese School of Economics Universy of New England Armidale, NSW 2351 Australia Running Tle: Metafrontier Framewors Keywords: Stochastic Frontier Analysis, Data Envelopment Analysis, Technical Efficiency 1 Corresponding author.

2 Metafrontier Framewors for the Study of Firm-Level Efficiencies and Technology Ratios Christopher J. O Donnell, D.S. Prasada Rao, George E. Battese Abstract. This paper uses the concept of a metafrontier to compare the technical efficiencies of firms that may be classified into different groups. The paper presents the basic analytical framewor necessary for the definion of a metafrontier, shows how a metafrontier can be estimated using non-parametric and parametric methods, and presents an empirical application using cross-country agricultural sector data. The paper also explores the issues of technological change, time-varying technical inefficiency, multiple outputs, different efficiency orientations, and firm heterogeney. The authors than the edor, associate edor and the referees for comments and suggestions that have contributed to improving the paper. 2

3 1. Introduction Firms in different industries, regions and/or countries 2 face different production opportunies. Technically, they mae choices from different sets of feasible inputoutput combinations. These so-called technology sets differ because of differences in available stocs of physical, human and financial capal (e.g., type of machinery, size and qualy of the labour force, access to foreign exchange), economic infrastructure (e.g., number of ports, access to marets), resource endowments (e.g., qualy of soils, climate, energy resources) and any other characteristics of the physical, social and economic environment in which production taes place. Such differences have led efficiency researchers to estimate separate production frontiers for different groups of firms. For example, separate frontiers have been estimated for universies in Canada (McMillan and Chan, 2004), Australia (Worthington and Lee, 2005) and the Uned Kingdom (Glass, McKillop and Hyndman, 1995), and for ban branches in South Africa (O Donnell and van der Westhuizen, 2002) and Spain (Lovell and Pastor, 1997). After using data on a group of firms to estimate a production frontier, is common and straightforward to measure the relative performance of firms whin the group (e.g., Canadian universies). However, there is often considerable interest in measuring the performance of firms across groups (e.g., comparing efficiency levels in Canadian universies wh efficiency levels in Australian universies). Unfortunately, such comparisons are only meaningful in the liming special case where frontiers for different groups of firms are identical. As a general rule, efficiency levels measured relative to one frontier (e.g., the Canadian frontier) cannot be compared wh efficiency levels measured relative to another frontier (the Australian frontier). One of the aims of this paper is to develop a formal theoretical framewor for maing efficiency comparisons across groups of firms. We do this by measuring efficiency 2 In this paper, we use the term firm in a generic sense (i.e., to refer to all types of production enties and decision-maing uns) and we restrict our attention to cross-sectional variations in technology sets (i.e., variations across industries and countries). The concepts and methods we discuss are also relevant and valid when technology sets vary over time. 3

4 relative to a common metafrontier, defined as the boundary of an unrestricted technology set. We also define group frontiers to be the boundaries of restricted technology sets, where the restrictions derive from lac of economic infrastructure and/or other characteristics of the production environment, as discussed above. Importantly, the metafrontier envelops the group frontiers. 3 Thus, efficiencies measured relative to the metafrontier can be decomposed into two components: a component that measures the distance from an input-output point to the group frontier (the common measure of technical efficiency); and a component that measures the distance between the group frontier and the metafrontier (representing the restrictive nature of the production environment). Estimates of technical efficiency are often used to design programs for performance improvement. These programs involve changes to the management and structure of the firm. Estimates of the gap between group frontiers and the metafrontier can also be used to design programs for performance improvement, but these programs involve changes to the production environment. Governments can change characteristics of the production environment by, for example, building roads and ports, deregulating financial marets, and relaxing labour laws. Firms in some industries (e.g., finance and manufacturing) can also change characteristics of their production environment through relocation (wness the number of U.S. clothing firms that have relocated their manufacturing operations to south-east Asia, and the number of telephone call centres that now operate out of India). Governments have reduced capacy to change some of the physical and cultural characteristics of the production environment (e.g., soil qualy), and firms in some industries (e.g., agriculture) may have limed capacy to move their operations to more productive locations. In such cases, measures of the gaps between group frontiers and the metafrontier are informative, but may be of ltle use in designing performance-improvement programs. Another aim of this paper is to show how metafrontiers and group frontiers can be estimated using data envelopment analysis (DEA) and stochastic frontier analysis 3 Thus, the metafrontier is related to the concept of the metaproduction function defined by Hayami and Ruttan (1971, p. 82): the metaproduction function can be regarded as the envelope of commonly conceived neoclassical production functions. 4

5 (SFA) techniques. Battese and Rao (2002) present an SFA approach to the estimation of metafrontiers that is implicly underpinned by two different data-generating mechanisms, one that explains deviations between observed outputs and (fixed) group frontiers, and another that explains deviations between observed outputs and the metafrontier (also fixed). The problem wh this approach is that points on the estimated metafrontier may lie below points on the estimated group frontiers. Battese, Rao and O Donnell (2004) solve the problem by specifying a single data-generating process that explains deviations between observed outputs and group frontiers, and by defining the metafrontier to be a function that envelops the deterministic components of the group frontiers. However, they only consider estimation of the metafrontier using one type of SFA methodology. This paper considers both DEA and (alternative) SFA approaches to estimating both metafrontiers and group frontiers, and for decomposing differences in performance into technical efficiency and technology gap effects. To highlight differences between the DEA and SFA approaches, we use country-level data drawn from the Food and Agriculture Organization (FAO) of the Uned Nations to mae inter-regional comparisons of agricultural efficiency. The final aim of the paper is to confirm that our basic analytical framewor and estimation methods can be extended to deal wh issues such as technological change, time-varying inefficiency effects, and multiple outputs. The abily to deal wh these issues means that the metafrontier approach can be adapted for use in empirical contexts where relatively sophisticated DEA and SFA methods are already routinely applied. The structure of the paper is as follows. In Section 2, we explain the microeconomic theory underpinning metafrontiers, and show how distances between observed data points and the metafrontier can be decomposed into what we refer to as metatechnology ratios and conventional radial measures of technical efficiency. In Section 3, we explain how group frontiers and metafrontiers can be estimated using DEA and SFA methods. SFA estimation is especially complicated by the theoretical requirement that the metafrontier envelops the group frontiers. In Section 4, we illustrate the differences between the DEA and SFA approaches by estimating agricultural production frontiers for countries in Africa, the Americas, Europe and Asia (i.e., four groups). In Section 5, we discuss extensions to the methodological 5

6 framewor to accommodate technological change, time-invariant inefficiency effects, multiple outputs, and alternative efficiency orientations. In Section 6, we summarise the paper and identify two opportunies for further research. 2. The Basic Analytical Framewor Efficiency measurement is deeply rooted in production theory and the concept of distance functions. In this section, we define the metafrontier and group frontiers in terms of output sets and output distance functions. We then show how output distance functions can be used to define technical efficiencies and metatechnology ratios. 2.1 The Metafrontier Let y and x be nonnegative real output and input vectors of dimension M 1 and N 1, respectively. The metatechnology set contains all input-output combinations that are technologically feasible. Formally: {(, ) : ; ; can produce } T = xy x 0y 0 x y. (1) Associated wh this metatechnology set are input and output sets. For example, the output set is defined for any input vector, x, as: { T} P( x) = y:( x, y ). (2) We refer to the boundary of this output set as the output metafrontier. We assume the output set satisfies the standard regulary properties listed in Färe and Primont (1995). Since the main focus of this paper is to measure efficiency, is convenient to represent the technology using the output metadistance function, defined as: { P } D( xy, ) = infθ θ > 0:( y/ θ) ( x ). (3) 6

7 This function gives the maximum amount by which a firm can radially expand s output vector, given an input vector. The distance function inhers s regulary properties from the regulary properties of the output set. An observation (x, y) can be considered technically efficient wh respect to the metafrontier if and only if D ( xy, ) = Group Frontiers It is also possible to conceptualise the existence of sub-technologies that represent the production possibilies of groups of firms. We consider the case where the universe of firms can be divided into K (>1) groups, and we suppose that resource, regulatory or other environmental constraints may prevent firms in certain groups from choosing from the full range of technologically feasible input-output combinations in the metatechnology set, T. Rather, the input-output combinations available to firms in the -th group are contained in the group-specific technology set: T {(, ) : ; ; can be used by firms in group to produce } = xy x 0y 0 x y. (4) The K group-specific technologies can also be represented by the following groupspecific output sets and output distance functions: { T } P ( x) = y:( x, y ), = 1, 2,, K; and (5) D { P } ( xy, ) = infθ θ > 0:( y/ θ) ( x ), = 1, 2,, K. (6) We refer to the boundaries of the group-specific output sets as group frontiers. If the output sets, P ( x ), = 1, 2,, K, satisfy standard regulary properties then the distance functions, D ( xy, ), = 1, 2,, K, also satisfy standard regulary properties. Irrespective of the properties of these sets and functions, is clear that R.1 If ( xy, ) T for any then ( xy, ) T ; R.2 If ( xy, ) T then ( xy, ) T for some ; 7

8 1 2 K R.3 T { T T... T } = ; and R.4 D ( xy, ) D( xy, ) for all = 1, 2,, K. These properties follow from the fact that the group-specific output sets, P ( x ), = 1, 2,, K, are subsets of the unrestricted output set, P( x ). This is illustrated in Figure 1 4 where we depict the production possibilies available to single-input, single-output firms from three different groups. The group- frontier is labelled - and is assumed to be convex ( = 1, 2, 3). If the three groups are exhaustive (i.e., if K = 3) then the group-specific frontiers envelop all the input-output combinations that could be produced by any single firm, implying the metafrontier is the nonconvex piecewise frontier, 1-B-3. However, if the three groups are not exhaustive, then other input-output combinations may be feasible and the metafrontier could conceivably be the convex frontier, M-M. Thus, R.5 Convex P(x) does not necessarily imply convex group output sets, P (x), = 1, 2,, K; and vice versa. [FIGURE 1 NEAR HERE] 2.3 Technical Efficiencies and Metatechnology Ratios Recall from Section 2.1 that an observation (x, y) is technically efficient wh respect to the metafrontier if and only if D ( xy, ) = 1. More generally, an output-orientated measure of the technical efficiency of an observed pair (x, y) wh respect to the metatechnology is: TE( xy, ) = D( xy, ). (7) 4 We than Gunnar Breustedt from the Department of Agricultural Economics at the Universy of Kiel for his comments made on the metafrontier, as proposed in Battese, Rao and O Donnell (2004). These comments, which were made in a personal communication to George Battese, have influenced our discussion of Figure 1. 8

9 For example, D ( xy, ) = 0.6 indicates that the output vector, y, is 60 per cent of the maximum output that could be produced by a firm using the input vector, x. We can also measure technical efficiency wh respect to the group- frontier. Specifically, an output-orientated measure of technical efficiency wh respect to the technology of group is: TE ( xy, ) = D ( xy, ). (8) For example, if D ( xy, ) = 0.8 then output is 80 per cent of the maximum output that could be produced by a firm using the input vector, x, and group- technology. Result R.4 states that the group- output distance function, D ( xy, ), can tae a value no less than the output metadistance function, D( xy, ). This is another way of saying that the metafrontier envelops the group- frontier. Whenever a strict inequaly is observed between the group- distance function and the metadistance function, we can obtain a measure of how close the group- frontier is to the metafrontier. Specifically, the output-orientated metatechnology ratio 5 for group- firms is defined as: D( xy, ) TE( xy, ) MTR ( xy, ) = =. D ( xy, ) TE ( xy, ) (9) Using the numerical examples above, where the technical efficiency of (x, y) wh respect to the metatechnology was 0.6 and the technical efficiency wh respect to the group- frontier was 0.8, the metatechnology ratio is found to be 0.75 (= 0.6/0.8). This means that, given the input vector, the maximum output that could be produced by a firm from group is 75 per cent of the output that is feasible using the metatechnology. 5 Battese, Rao and O Donnell (2004, p. 94) refer to this measure as the technology gap ratio. However, increases in the (technology gap) ratio imply decreases in the gap between the group frontier and the metafrontier. To avoid confusion, we use the term metatechnology ratio here. 9

10 Equation (9) provides for the following convenient decomposion of the technical efficiency of a particular input-output combination: TE( xy, ) = TE ( xy, ) MTR ( xy, ). (10) This shows that technical efficiency measured wh reference to the metafrontier (representing the existing state of nowledge) can be decomposed into the product of technical efficiency measured wh reference to the group- frontier (representing the existing state of nowledge and the physical, social and economic environment that characterises group ) and the metatechnology ratio for group- (which measures how close the group- frontier is to the metafrontier). Policies and programs for efficiency improvement are often targeted at eher firms (e.g., education and training programs) or characteristics of the firm s operating environment (e.g., construction of railways and port facilies). The decomposion given by equation (10) is useful because allows policymaers to better assess the potential payoffs from these different types of programs. Finally, the decision to assume a convex or nonconvex metafrontier has implications for measures of efficiency and the metatechnology ratio. For example, consider a firm from group 2 that produces at the input-output combination labelled A in Figure 1. If the metafrontier is the nonconvex frontier, 1-B-3, then our measures of technical efficiency and the metatechnology ratio are 0C TE( A) = 0.67, (11) 0E 2 0C TE ( A) = 0.74, and (12) 0D 2 TE( A) 0 C /0E 0D MTR ( A) = = = (13) 2 TE ( A) 0 C /0D 0E However, if the metafrontier is the convex function labelled M-M, our measures are: 10

11 0C TE( A) = 0.60, (14) 0F 2 0C TE ( A) = 0.74, and (15) 0D 2 TE( A) 0 C /0F 0D MTR ( A) = = = (16) 2 TE ( A) 0 C /0D 0F Thus, the convexy property leads to lower metatechnology ratios and lower measures of efficiency wh respect to the metafrontier. 3. Estimation Empirical measurement of technical efficiencies and metatechnology ratios requires an empirical description of the technology. Once data on the inputs and outputs for random samples of firms from the different groups are available, we can estimate the metafrontier and the group frontiers using DEA or SFA. In this section, we consider the problem of estimating the metafrontier when panel data are available on singleoutput firms. The case of multiple-output firms is discussed in Section Data Envelopment Analysis Two types of DEA models are found in the mainstream efficiency lerature an input-orientated model and an output-orientated model. In the input-orientated case, DEA defines the frontier by holding output levels constant and seeing the maximum proportional reduction in input usage that is compatible wh the technology set. In the output-orientated case, the DEA method holds inputs constant and sees the maximum possible proportional increase in outputs. The two measures give the same technical efficiency scores if the technology exhibs constant returns-to-scale (CRS), but different scores when the technology exhibs variable returns-to-scale (VRS). 11

12 It is possible to construct a convex group- frontier by applying the DEA method to all the observed inputs and outputs of firms in that group. If group consists of data on L firms and there are T periods, the VRS output-orientated DEA problem is: max φ, λ φ s.t. φ y y λ 0, Xλ x 0, j λ = 1 and λ 0. (17) where y is the output quanty for the i-th firm in the t-th period; x is the N 1 vector of input quanties for the i-th firm in the t-th period; y is the L T 1 vector of output quanties for all L firms in all T periods; X is the N L T matrix of input quanties for all L firms in all T periods; j is an L T 1 vector of ones; λ is an L T 1 vector of weights; and φ is a scalar. The value of φ that solves the linear program (LP) defined by equation (17) can be shown to be no less than one and provides information on the technical efficiency of the i-th firm in the t-th period. Specifically, φ 1 is the proportional increase in outputs that could be achieved when the input quanties of the i-th firm in the t-th period are held constant. Thus, 1/ φ is (an estimate of) the output-orientated technical efficiency measure given by equation (8). The value of λ that solves the LP defined by equation (17) provides information on the peers of the i-th firm in the t-th period. These peers are efficient points that define the facet of the frontier onto which the inputs and outputs of the i-th firm in the t-th period are projected. Solving the LP defined by equation (17) separately for every firm in the group in every time period identifies all the facets on the group- frontier. In practice, the tedious wor of solving 12

13 a different LP for every firm in every period is usually undertaen using purpose-built software pacages such as DEAP (see Coelli, 1996b). A convex metafrontier can be identified by applying the DEA method to the inputs and outputs of all L = L firms in all T periods. The structure of the metafrontier LP is identical to that of equation (17) except that X is of dimension N LT, and y and λ are LT 1. Solving this metafrontier LP separately for every firm in the sample in every period yields estimates of firm efficiencies wh respect to the metafrontier, and also identifies all the facets on the metafrontier. The value of φ that solves the group problem can be shown to be no greater than the value of φ that solves the metafrontier problem. 6 Thus, firms will be no more technically efficient when they are assessed against the metafrontier than against the group frontiers, and the metafrontier will never lie below any of the group frontiers. Finally, having estimated the technical efficiencies of firms wh respect to the metafrontier and group frontiers, is straightforward to estimate the metatechnology ratio at observed input and output levels using equation (9). 3.2 Stochastic Frontier Analysis SFA involves parameterising the frontier and estimating using econometric techniques. A stochastic group- frontier model is V U y = f( x, x,..., x ; β ) e (18) 1 2 N where x n is the n-th input quanty of the i-th firm in the t-th period; β is an unnown parameter vector associated wh the -th group; the V s represent statistical noise and are assumed to be independently and identically distributed as N σ -random variables; and the U s represent inefficiency and are defined by 2 (0, v ) 6 The dual to the group- maximisation LP is a minimisation problem wh L T+1 constraints, while the dual to the metafrontier maximisation LP is a minimisation problem wh LT+1 constraints. The constraints in the group- dual problem are a subset of the constraints in the metafrontier problem, so the minimised value of the former is no greater than the minimised value of the latter. 13

14 the truncation (at zero) of N μ σ -distributions, where the μ s are defined by 2 (, ) some appropriate inefficiency model (e.g., the model of Battese and Coelli, 1995) 7. If the exponent of the frontier production function is linear in the parameter vector, then the model can be wrten β, V U V U y f( x, x,..., x ; β ) e e β + = x (19) 1 2 N where x is now a vector of (transformations of) inputs for the i-th firm in the t-th period. Data on the inputs and outputs of firms in the -th group can be used to obtain eher least-squares or maximum-lielihood (ML) estimates of the unnown parameters of this frontier. ML estimates are programmed to be calculated in the software pacage FRONTIER (Coelli, 1996a). Following estimation, the technical efficiency of the i-th firm in the t-th period wh respect to the group- frontier can be obtained using the result: TE y = = e x + V e β U. (20) A predictor for this technical efficiency measure, as proposed in Battese and Coelli (1988), is also programmed to be calculated in FRONTIER. A deterministic metafrontier production function is y = f( x, x,..., x ; β ) e x β (21) * 1 2 N where * y is the metafrontier output and β is a vector of metafrontier parameters satisfying the constraints xβ x β for all = 1, 2,, K. (22) 7 An alternative model that accounts for heteroscedasticy is proposed by Wang (2002). 14

15 Two features of the model given by equations (19) to (22) are noteworthy. First, the constraints given by (22) imply that the metafrontier function cannot fall below any of the group frontiers. Second, the model is underpinned by only one data-generating process. This contrasts wh the stochastic metafrontier model of Battese and Rao (2002) that assumes a different data-generating mechanism for the metafrontier than for the different group frontiers. Their stochastic metafrontier can be conveniently estimated using FRONTIER (using the inputs and outputs of all firms in all groups in all time periods), but there is no guarantee that the estimated metafrontier will envelop the estimated group frontiers. An estimated metafrontier function that envelops the estimated group frontiers can be obtained by solving the optimisation problem: min L T ln (,,..., ; ) ln (,,..., ; ˆ f x x x β f x x x β ) 1 2 N 1 2 N β i= 1 t= 1 ln (,,..., ; ) ln (,,..., ; ˆ β ), for all i and t; (23) s.t. f x1 x2 xn β f x1 x2 xn where ˆ β is the estimated coefficient vector associated wh the group- stochastic frontier. Since these estimated coefficient vectors are fixed for the above problem, an equivalent form of the LP defined by equation (23) is min L T ln f( x, x,..., x ; β ) 1 2 N β i= 1 t= 1 ln (,,..., ; ) ln (,,..., ; ˆ β ), for all i and t. (24) s.t. f x1 x2 xn β f x1 x2 xn Furthermore, if the function f(.) is log-linear in the parameters (as assumed in the empirical application in this paper), the LP problem becomes: min β s.t. x β ˆ β x β x for all i and t, (25) where x is the arhmetic average of the x -vectors over all firms in all periods. 15

16 This optimisation problem, and a similar problem involving minimisation of a sum of squared deviations, is discussed in more detail in Battese, Rao and O Donnell (2004). Standard errors for the estimators for the metafrontier parameters can be obtained using simulation or bootstrapping methods. After solving the LP problem defined by equation (25), estimates of metatechnology ratios and technical efficiencies wh respect to the metafrontier can be obtained using the following decomposion of equation (19): x β U e x β + V y = e e. (26) x β e The first term on the right-hand side is the technical efficiency of the i-th firm in the t-th period wh respect to the group- frontier, defined by equation (20). The second term on the right-hand side is the metatechnology ratio for the i-th firm in the t-th period (in the -th group): MTR x β e =. (27) x β e Estimating the metatechnology ratio is simply a matter of substuting estimates of β and β into equation (27). The constraints in the LP problem defined by equation (25) guarantee that metatechnology ratios estimated in this manner will lie in the un interval. Finally, equations (9), (20), (26) and (27) together imply that the technical efficiency of the i-th firm in the t-th period wh respect to the metafrontier is TE y = x. (28) + V e β Thus, technical efficiency relative to the metafrontier is defined in an analogous way to equation (20) is the ratio of the observed output relative to the frontier output, adjusted for the corresponding random error. In practice, is convenient to predict technical efficiency wh respect to the metafrontier using the decomposion: 16

17 ˆ ˆ ˆ TE = TE MTR (29) where and (27). TE ˆ and MTR ˆ are the predictors discussed in connection wh equations (20) 4. Empirical Application In this section, we use country-level data drawn from the Food and Agriculture Organization (FAO) of the Uned Nations to mae inter-regional comparisons of agricultural efficiency. Our main aim is not to provide a detailed analysis of global agricultural productivy, but rather to illustrate the concepts and methods discussed in Sections 2 and 3 above. Because our application is illustrative, we mae several simplifying assumptions concerning, for example, technological change and the timevarying nature of technical inefficiency. Methods for relaxing some of these assumptions are discussed in Section Data We use data exclusively drawn from the FAOSTAT system used for dissemination of statistics compiled at the Statistics Division of the FAO in Rome. The Statistics Division maintains a webse where data on agriculture are made available to potential users. The se can be reached through the web address: The data set contains observations on 97 countries covering the five-year period from 1986 to These countries are evenly distributed across the globe and account for roughly 99 per cent of the world s population and agricultural output. To maintain the terminology of previous sections, the 97 countries in the data set can be thought of as firms, and these countries/firms are classified into four groups: Africa (27 countries); The Americas (21 countries); Asia (26 countries); and Europe (23 countries). These regional groupings are those defined by the FAO for s purposes of dissemination of information and data. We are not seeing to justify this grouping of the countries for a detailed analysis of agricultural productivy or efficiency. A complete list of the 17

18 countries in each group is provided in Table 1. Australia, New Zealand and Papua New Guinea are included in Asia. The former USSR is included in Europe. The data set comprises observations on one output variable (an aggregate of 185 agricultural commodies) and five input variables 8 (land, machinery, labour, fertiliser and livestoc). A full description of the variables in our empirical application is provided in Appendix A. 4.2 DEA Results DEA estimates of the group frontiers and the metafrontier were obtained using DEAP 2.1 (see Coelli, 1996b). All results were obtained using the VRS output-orientated DEA model given by equation (17). In the remainder of this paper, the acronyms DEA-K and DEA-MF are used to refer to DEA estimates of technical efficiencies relative to the group- frontiers and the metafrontier, respectively. The acronym DEA-MTR is used to refer to DEA estimates of the metatechnology ratio. Technical efficiencies and metatechnology ratios were estimated for each of the 97 countries in each of five years. Table 2 reports descriptive statistics for these estimates for selected countries, for the four groups, and for all countries combined. For example, the first row reports that the technical efficiency of Argentina wh respect to the group frontier was estimated to vary between and during the five-year period, wh an average value of and a standard deviation of The DEA-K results reveal that the UK, for example, was about 97% efficient when measured relative to the European frontier. This means that average agricultural output in the UK is approximately 97% of the output that is possible using the same input levels and the production technology available in Europe. The DEA-MTR results reveal that the average metatechnology ratio for the UK is about This means that the maximum output that could be produced using the inputs of the UK and the technology of Europe is about 94% of the maximum output that could be produced using the same inputs and the technology represented by the metafrontier. 8 We restrict the number of outputs and inputs used in order to maintain degrees of freedom for details, see Coelli et al (2005). 18

19 Observe from Table 2 that the technical efficiency of South Africa is que high when measured wh respect to the African frontier (0.964) but low when measured against the metafrontier (0.610). This difference implies a low metatechnology ratio. Indeed, the average DEA-MTR estimate for South Africa is 0.633, indicating that the maximum output that is feasible using the African technology (and the input levels used by South Africa) is only about 63% of the output that could be achieved using the technology represented by the metafrontier. The results reported in Table 2 for different groups of countries suggest that, on average, African countries produce agricultural outputs under condions that are more restrictive than in other regions of the world the average metatechnology ratio (0.886) suggests that African countries could, at best, produce only 89% of the agricultural output that could be produced using the (unrestricted) metatechnology. The average metatechnology ratio for Asia (0.925) suggests that Asian countries could produce about 93% of the output that could be produced using the metatechnology (and the same inputs). Finally, is worth noting that many of the DEA-MF maximum values reported in Table 2 are equal to 1. For these countries, there must have been at least one year when they used an input-output combination that placed them at the point of tangency between their group frontiers and the metafrontier. 9 However, in the case of South Africa, for which we report a maximum DEA-K estimate of 1 and a maximum DEA- MTR estimate of 0.655, there must have been at least one year when used an inputoutput combination that placed on the African frontier, but well below the metafrontier. 4.3 SFA Results SFA estimates of the frontier models defined by equations (19) and (21) were obtained by assuming a translog functional form. The region- frontier is defined by 9 More precisely, since the metafrontier and group frontiers are formed as the intersection of several hyperplanes, there was at least one year when each of these countries operated at a point where the hyperplane of their group frontiers touched a hyperplane of the metafrontier. The results for the four groups of countries suggest that there was at least one country in each of the groups that operated at such a point. 19

20 5 5 5 = 0 + j j + jm j m + j= 1 j= 1 m= 1 ln y β β ln x 0.5 β (ln x )(ln x ) V U (30) where β = β for all j and m. For simplicy, we assumed U jm mj = U where the U s i i are half-normal random variables. Zero observations in the data set were handled using the approach suggested by Battese (1997). 10 SFA estimates of the parameters of the group frontiers were obtained using FRONTIER 4.1c (see Coelli, 1996a), while the parameters of the metafrontier were estimated using the SHAZAM code that is presented in Appendix B. SFA estimates of technical efficiencies and metatechnology ratios are summarised in Table 3. In this table, we use the acronyms SFA-K and SFA-MF to refer to technical efficiencies relative to the estimated group- frontiers and the estimated metafrontier, respectively. We also use the acronym SFA-POOL to refer to technical efficiencies obtained using the pooled model of Battese and Rao (2002), discussed in Section 3.2. Estimating this pooled model allows us to formally test for differences between the group frontiers. Specifically, the generalised lielihood-ratio test statistic for the null hypothesis that the group frontiers are identical is LR = This has a p-value of (using a chi-square distribution wh 69 degrees of freedom), so we reject the null hypothesis that the group frontiers are the same. This implies that the use of a metafrontier framewor is appropriate. The SFA efficiency estimates reported in Table 3 are generally lower than the DEA estimates reported in Table 2. For the ten selected countries listed in the tables, the SFA estimates for the group frontiers are lower than the DEA estimates for eight of the ten countries. Thus, DEA seems to f a tighter frontier than does SFA, as does in most empirical applications. 11 Countries in the Americas are still estimated to be the most technically efficient when assessed against both the group frontier and the 10 Two of the 130 observations in Asia (Group 3) had zero values for the fertiliser input, so for these observations the logarhm of the fertiliser variable is undefined. The Battese (1997) solution is to specify a different production frontier for these observations. Thus, for observations wh zero values for the fertiliser input, we use a dummy variable to both remove any terms involving ln x 4 i and allow for a different intercept. 11 The DEA results on efficiency are probably misleading for an agricultural application in which random errors are highly significant, even though the technical inefficiency effects seem to be dominant. 20

21 metafrontier. However, the metatechnology ratio for Africa is now estimated to be higher than that of any other group (i.e., Africa is now found to be the least restrictive production environment). There are some important differences between the SFA and DEA estimates of technical efficiencies and metatechnology ratios for selected countries. For example, Indonesia is now estimated to be relatively inefficient wh respect to the Asian frontier (s average efficiency score is using SFA compared wh using DEA), and the Indonesian metatechnology ratio is now que low (0.830 compared wh ). These results suggest that the SFA estimate of the Asian frontier is considerably higher than the DEA estimate in the region of the input-output combinations used by Indonesia. 12 Several of the estimated metatechnology ratios summarised in Table 3 have maximum values of 1. This means that several countries could have used their inputs to place themselves at points of tangency between the group frontiers and the metafrontier. Countries in this posion (from among the ten selected countries) were Argentina, Australia, China, the Netherlands and the USA. Finally, is worth noting that there are some significant differences between the SFA-MF efficiency estimates and the SFA-POOL efficiency estimates, the latter having being obtained using the stochastic metafrontier model of Battese and Rao (2002). In many cases (e.g., Australia, China and the Netherlands), the SFA-MF estimates (i.e., 0.921, and 0.932) are more plausible than the SFA-POOL estimates (0.347, and 0.351). 5. Extensions to the Basic Framewor Sections 3 and 4 describe and illustrate basic non-parametric and parametric methods for estimating single-output group frontiers and the metafrontier using panel data. The recent efficiency lerature explores several extensions to conventional non-parametric 12 By a similar line of reasoning, the two estimated frontiers are much closer to each other in the region of the input-output combinations used by, for example, Australia. 21

22 and parametric frontier models and estimators, and, in this section, we consider some of these extensions in the context of the metafrontier. 5.1 Technological Change The group frontiers and the metafrontier depicted in Figure 1 are boundaries of restricted and unrestricted production possibilies sets. Technological change generally causes one or more of these sets to expand (i.e., leads to an increase in the number of feasible input-output combinations), implying an outward movement in one or more group frontiers and/or the metafrontier. To allow for this possibily, we must modify our metafrontier estimation strategy. In the case of DEA, we would first solve the LP problem of equation (17) using only observations from period 1 (this yields an estimate of the period-1 frontier), then solve the problem again using observations from periods 1 and 2 (this yields a period-2 frontier that envelops the period-1 frontier), and so on until we eventually solve the problem using all the observations in the data set (this yields a period-t frontier that envelops the frontiers from all earlier periods). In the case of SFA, we would simply include a time trend in the deterministic component of the model details can be found in Coelli et al. (2005). 5.2 Time-Invariant Inefficiency Effects The SFA model defined by equation (18) and the DEA problem given by equation (17) both allow for variations in technical inefficiency effects over time. If we assume U = U then the SFA model yields time-invariant estimates of technical efficiencies i wh respect to the group frontiers (this simplifying assumption underpinned the SFA results in Section 4). However, estimates of metatechnology ratios and technical efficiencies wh respect to the metafrontier are still time-varying. Unfortunately, the DEA problem defined by equation (17) cannot be modified to yield meaningful time-invariant technical efficiency estimates. When time-invariant DEA estimates are required, is a common practice to simply tae averages of the timevarying estimates. 22

23 5.3 Multiple-Outputs The multiple-output version of the DEA problem defined by (17) is: max φ, λ φ s.t. φ y Yλ 0, Xλ x 0, j λ = 1 and λ 0. (31) where y is an M 1 vector of output quanties for the i-th firm in the t-th period; Y is the M L T matrix of output quanties for all L firms in all T periods; and all other scalars, vectors and matrices are exactly as they are defined in Section 3.1. Importantly, 1/ φ is still an output-orientated technical efficiency measure wh respect to the group frontier or the metafrontier (depending on whether the LP is solved using the group data or all the data in the sample), and the metatechnology ratio can still be estimated using equation (9). The multiple-output analogue of the single-output stochastic production frontier is a stochastic distance function. For example, a stochastic translog version of the group- output distance function is: N N N ln ym = β 0 + βjln x j β js( ln x j )( ln xs ) j= 1 j= 1s= 1 M 1 M 1M 1 ( y y ) α ( y y ) ( y y ) + α ln ln ln j m M mn m M n M m= 1 m= 1 n= 1 N M 1 γ mn ln x j ln( yn / ym ) + V + U (32) j= 1 n= 1 where U = ln D ( x1, x2,..., x, y1, y2,..., y ) captures inefficiency. This model N M can be estimated using standard SFA methods, although there is debate in the lerature concerning endogeney bias for details, see Atinson, Färe and Primont 23

24 (1998) and Coelli (2000). Following estimation of the group- distance functions, an estimate of the metafrontier can be obtained by solving a generalisation of the optimisation problem defined by equation (23). 5.4 Alternative Orientations The metafrontier approach discussed in this paper can accommodate alternative efficiency orientations. The input-orientated analogue of the VRS multiple-output DEA problem defined by equation (31) is given by: min φ, λ φ s.t. Yλ y 0, φ x Xλ 0, j λ = 1 and λ 0. (33) Recall from Section 3.1 that the input-orientated and output-orientated problems yield identical estimates of technical efficiency under the assumption of constant returns to scale. To impose the CRS property on the estimated DEA frontier we simply remove the constraint that the λ s sum to one. An input-orientated analogue of the group- output distance function in equation (32) is also available for use in a metafrontier framewor. Both types of distance functions are special cases of the directional distance function discussed by Chambers, Chung and Färe (1996). 13 Directional distance functions are particularly useful for measuring the efficiency of firms that produce both good and bad outputs. For example, Färe et al. (2005) use a quadratic directional distance function to measure the technical efficiencies of US electric utilies (producers of electricy and sulphur dioxide). 13 The directional distance function is a variant of the shortage function of Luenberger (1992, 1995). 24

25 5.5 Identifying Groups In most practical settings, firms can be grouped a priori on the basis of geographical, economic and/or polical boundaries. In the absence of such boundaries, multivariate statistical techniques (e.g., cluster analysis) are available for determining numbers of groups and group membership. When the analysis is conducted in an SFA framewor, is also possible to conduct statistical tests concerning the number of groups. For example, when the SFA model defined by equation (19) is employed, we can test whether groups j and l should be amalgamated into a single group by testing the joint null hypothesis : j l H 0 β = β for all m= 0,1,..., K'. The LR test reported in Section 4.2 is a more general test of this type is a test of the null hypothesis that all (four) groups of firms (countries) can be amalgamated into a single group. m m More generally, statistical methods for determining and/or confirming the number of groups (and group membership) range from estimating and testing the coefficients of dummy-variable models, to estimating and testing the parameters of mixtures models (e.g., Orea and Kumbhaar, 2004; O Donnell and Griffhs, 2006). 6. Conclusion This paper develops the concept of the metafrontier for the purpose of studying differences in efficiency across groups of firms. The metafrontier is defined as the boundary of an unrestricted technology set. Groups of firms that operate in resourcepoor or highly-regulated production environments may only have access to a restricted technology set. We refer to the boundaries of these restricted sets as group frontiers. In this paper, the closeness of group frontiers to the metafrontier are measured as metatechnology ratios for the different groups. These ratios are of considerable interest to managers and government policy-maers, not least because they measure the potential improvement in performance resulting from changes in the production 25

26 environment. Governments can change the production environment by investing in physical, financial and human capal (e.g., by building ports and power stations, creating cred marets, or by investing in education and training); managers may have the capacy to change the production environment by moving their operations from one location to another. The paper shows how group frontiers and the metafrontier can be estimated using DEA and SFA methodologies. Both approaches are popular in the efficiency lerature. An empirical example using cross-country agricultural data provides evidence that agricultural producers in Europe operate in relatively restrictive production environments (using the SFA results). Our empirical example illustrates the use of the DEA and SFA methodologies in a single-output context under various restrictive assumptions concerning the nature of the technology and firm inefficiency. However, the concepts and methods developed in the paper readily extend to multiple-output firms and models that are based on different assumptions concerning inefficiency and technological change. In conclusion, we mention two opportunies for improving of our SFA estimator. First, observe that the linear program used to construct the SFA metafrontier (i.e., equation 23) gives equal weight to all firms, irrespective of how close they may be to their respective group frontiers. This is in contrast to the linear program used to construct the DEA metafrontier, which can be shown to give fully-efficient firms a weight of one and inefficient firms a weight of zero. This general idea can be implemented in a stochastic frontier context by multiplying the term in the square bracets in equation (23) by TE ˆ. Second, econometric theory suggests that more efficient estimators of the parameters of both the group frontiers and the metafrontier can be obtained by estimating the group frontiers in a seemingly unrelated regression (SUR) framewor. The size of the efficiency gains is an empirical question that seems worthy of investigation. 26

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